You are making a wise decision to concentrate on Yellowstone NP and northwestern Wyoming after the Black Hills NP. Between the two national parks, you may have time to visit Devil's Tower NM and the Little Bighorn Battlefield. And don't miss the Buffalo Bill Museum in Cody, just outside the east gate of Yellowstone. Great art and gun collections, but much more than that for the whole family.
Just the Yellowstone-Grand Teton area has more than enough to keep your kids occupied. If the kids are pretty adventurous and not too tired of driving, if you don't mind driving on twisting mountain roads, and if you are in a motorhome with a toad or towing a trailer, plan to spend two nights in Cooke City, MT, while you are in Yellowstone. Then take your toad or tow vehicle (NOT a MH or trailer) on US 212 over Beartooth Pass to Red Lodge, then MT 72 and WY 296 back to Cooke City. The Beartooth Highway has been called "the most scenic drive in America." It is a road that your kids will love telling their friends in Ohio about when they get home!
If the kids are getting tired of camping by the time you leave Yellowstone, head for Colter Bay on Jackson Lake in Grand Teton NP. There you might still be able to reserve a log cabin with a big stone fireplace. Our kids loved building a fire in the fireplace and pretending they were Daniel Boone when they were that age. You can also rent a canoe at Colter Bay and paddle around near the shoreline, maybe catching a glimpse of a moose or an eagle in the early morning hours.
In the town of Jackson, you can take a raft trip on the Snake River, ride the tram to the top of the ski area, go horseback riding, or visit the National Museum of Wildlife Art. And, if you are tired of cooking in an RV by that point, there are plenty of great places to take the kids for meals.
Have fun in SD, WY, and MT and wait to experience Bryce and Zion on a trip just to Utah, Arizona, and Colorado. After all, Bryce and Zion are not the only sights you will want to see in that region. If you are driving that far, you don't want to miss Arches NP, Canyonlands NP, Dead Horse Point State Park, Capital Reef NP, Kodachrome SP, Coral Pink Sand Dunes SP, North Rim of Grand Canyon NP, Monument Valley, Hovenweep NM, and Mesa Verde NP. And you'll need at least three weeks for all that . . . after you get to Utah!